Category: Quant Finance

Optimal Labeling in Trading: Bridging the Gap Between Supervised and Reinforcement Learning

March 10, 2025 / Thijs van den Berg

When building trading strategies, a crucial decision is how to translate market information into trading actions.

Traditional supervised learning approaches tackle this by predicting price movements directly, essentially guessing if the price will move up or down.

Typically, we decide on labels in supervised learning by asking something like: “Will the price rise next week?” or “Will it increase more than 2% over the next few days?” While these are intuitive choices, they often seem arbitrarily tweaked and overlook the real implications on trading strategies. Choices like these silently influence trading frequency, transaction costs, risk exposure, and strategy performance, without clearly tying these outcomes to specific label modeling decisions. There’s a gap here between the supervised learning stage (forecasting) and the actual trading decisions, which resemble reinforcement learning actions.

In this post, I present a straightforward yet rigorous solution that bridges this gap, by formulating label selection itself as an optimization problem. Instead of guessing or relying on intuition, labels are derived from explicitly optimizing a defined trading performance objective -like returns or Sharpe ratio- while respecting realistic constraints such as transaction costs or position limits. The result is labeling that is no longer arbitrary, but transparently optimal and directly tied to trading performance.

Efficient Rolling Median with the Two-Heaps Algorithm. O(log n)

March 10, 2025 / Thijs van den Berg

Calculating the median of data points within a moving window is a common task in fields like finance, real-time analytics and signal processing. The main applications are anomal- and outlier-detection / removal.

Fig 1. A slow-moving signal with outlier-spikes (blue) and the rolling median filter (orange).

A naive implementation based on sorting is costly—especially for large window sizes. An elegant solution is the Two-Heaps Rolling Median algorithm, which maintains two balanced collections to quickly calculate the median as new data arrives and old data departs.

Fast Rolling Regression: An O(1) Sliding Window Implementation

March 10, 2025 / Thijs van den Berg

In finance and signal processing, detecting trends or smoothing noisy data streams efficiently is crucial. A popular tool for this task is a linear regression applied to a sliding (rolling) window of data points. This approach can serve as a low-pass filter or a trend detector, removing short-term fluctuations while preserving longer-term trends. However, naive methods for sliding-window regression can be computationally expensive, especially as the window grows larger, since their complexity typically scales with window size.

Fig 5. The impact of number of samples on the variance of the confidence distribution.

Understanding the Uncertainty of Correlation Estimates

March 9, 2025 / Thijs van den Berg

Correlation is everywhere in finance. It’s the backbone of portfolio optimization, risk management, and models like the CAPM. The idea is simple: mix assets that don’t move in sync, and you can reduce risk without sacrificing too much return. But there’s a problem—correlation is usually taken at face value, even though it’s often some form of an estimate based on historical data. …and that estimate comes with uncertainty!

This matters because small errors in correlation can throw off portfolio models. If you overestimate diversification, your portfolio might be riskier than expected. If you underestimate it, you could miss out on returns. In models like the CAPM, where correlation helps determine expected returns, bad estimates can lead to bad decisions.

Despite this, some asset managers don’t give much thought to how unstable correlation estimates can be. In this post, we’ll dig into the uncertainty behind empirical correlation, and how to quantify it.

Extracting Interest Rate Bounds from Option Prices

January 19, 2021 / Thijs van den Berg

In this post we describe a nice algorithm for computing implied interest rates upper- and lower-bounds from European option quotes. These bounds tell you what the highest and lowest effective interest rates are that you can get by depositing or borrowing risk-free money through combinations of option trades. Knowing these bounds allows you to do two things:

1. Compare implied interest rate levels in the option markets with other interest rate markets. If they don’t align then you do a combination of option trades to capture the difference.

2. Check if the best borrowing rate is higher than the lowest deposit rate. If this is not the case, then this means there is a tradable arbitrage opportunity in the market: you can trader a combination of options that effectively boils down to borrowing money at a certain rate, and at the same time depositing that money at a higher rate.

Recovering Accurate Implied Dividend and Interest Rate Term-Structures from Option Prices

January 4, 2021 / Thijs van den Berg

In this post we discuss the algorithms we use to accurately recover implied dividend and interest rates from option markets.

Implied dividends and interest rates show up in a wide variety of applications:

to link future-, call-, and put-prices together in a consistent market view
de-noise market (closing) prices of options and futures and stabilize PnL’s of option books
give tighter true bid-ask spreads based on parity and arbitrage relationships
compute accurate implied volatility smiles and surfaces
provide predictive models and trading strategies with signals based on implied dividends, and implied interest rate information

Validating Trading Backtests with Surrogate Time-Series

December 29, 2020 / Thijs van den Berg

Back-testing trading strategies is a dangerous business because there is a high risk you will keep tweaking your trading strategy model to make the back-test results better. When you do so, you’ll find out that after tweaking you have actually worsened the ‘live’ performance later on. The reason is that you’ve been overfitting your trading model to your back-test data through selection bias.

In this post we will use two techniques that help quantify and monitor the statistical significance of backtesting and tweaking:

First, we analyze the performance of backtest results by comparing them against random trading strategies that similar trading characteristics (time period, number of trades, long/short ratio). This quantifies specifically how “special” the timing of the trading strategy is while keeping all other things equal (like the trends, volatility, return distribution, and patterns in the traded asset).
Second, we analyse the impact and cost of tweaking strategies by comparing it against doing the same thing with random strategies. This allows us to see if improvements are significant, or simply what one would expect when picking the best strategy from a set of multiple variants.